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An accelerating universe has $\ddot{a}>0$, equivalently $q<0$ while expansion continues.

Here $a$ is the scale factor and $q=-a\ddot{a}/{\dot{a}}^2$ the deceleration parameter. Accelerating expansion means $\dot{a}>0$ and $\ddot{a}>0$; the Hubble parameter $H=\dot{a}/a$ can still decrease. The observational challenge is to reconstruct that history from a past light cone, rather than watch a distant galaxy speed up over the observing campaign.

For one conserved constant-$w$ perfect fluid in a flat expanding FLRW metric, $H(z)=H_0(1+z{)}^{3(1+w)/2}$. Integrating the radial null geodesic giveswhen $w\ne -1/3$. Its continuous coasting limit is $D_L=c(1+z)\log (1+z)/H_0$. Expansion gives $D_L=(c/H_0)[z+(1-3w)z^2/4+O(z^3)]$, agreeing with the deceleration parameter $q_0=(1+3w)/2$. Curvature, additional independently evolving fluids or a varying $w$ change this single-fluid expression.

For $V=V_0{e}^{-\lambda \phi /M_{\mathrm{Pl}}}$, $\lambda \ne 0$, the displayed ansatz satisfies the flat Friedmann equation and scalar field equation of motion when $V_0{t}_0^2=2M_{\mathrm{Pl}}^2(6-{\lambda }^2)/{\lambda }^4$. Its equation-of-state parameter is $w=-1+{\lambda }^2/3$ and its deceleration parameter is $q={\lambda }^2/2-1$. It gives exponential-potential power-law inflation only for $0<{\lambda }^2<2$; the same algebraic family includes decelerating expansion outside that range when its potential normalization is real.

For separately conserved pressureless matter, ${\rho }_m\propto a^{-3}$. The deceleration parameter identity $\dot{H}/H^2=-(1+q)$ and ${\Omega }_m\propto {\rho }_m/H^2$ give ${\dot{\Omega }}_m=H{\Omega }_m(2q-1)$. A dust-plus-vacuum universe has $q={\Omega }_m/2-{\Omega }_{\Lambda }$; conversion to conformal time yields the displayed flow. The familiar matter-only equation requires negligible cosmological constant. In a spatially flat dust-plus-vacuum universe, ${\Omega }_m+{\Omega }_{\Lambda }=1$ and ${\Omega }_m^{\prime }=-3\mathcal{H}{\Omega }_m{\Omega }_{\Lambda }$, so flatness does not require ${\Omega }_m=1$.

Type Ia supernovae are standardizable candles. Multi-epoch photometry measures peak brightness, light-curve shape and color; spectroscopy identifies the supernova type and supplies a redshift, preferably also using the host. Correcting the empirical width–luminosity relation and attenuation reduces the spread of absolute magnitudes. A nearby sample fixes a distance-ladder intercept, while distant events test the shape of the Hubble diagram. The observable relation isAn unknown common $M$ remains degenerate with the Hubble-constant intercept unless externally calibrated; relative distance-versus-redshift shape can still constrain acceleration.

For a cosmological spacetime described by an FLRW metric, the deceleration parameter is $q=-a\ddot{a}/{\dot{a}}^2$, and the low-redshift luminosity-distance expansion isA range of redshifts therefore tests the expansion history beyond the leading linear Hubble law. Within conventional matter-plus-cosmological-constant models, standardized distant supernovae fainter than a specified decelerating prediction imply larger luminosity distances and can favor $q_0<0$. For late-time matter and a cosmological constant, $q_0={\Omega }_{m,0}/2-{\Omega }_{\Lambda ,0}$. A flat $(0.3,0.7)$ example has $q_0=-0.55$. One brightness residual alone does not establish acceleration across arbitrary expansion histories or arbitrary attenuation models.

The supernova discovery of accelerating expansion combined repeated imaging, difference-image candidate searches, spectroscopic classification and calibrated light curves. In the High-Z survey, CCD discovery images and later host images separated transient light from its host; point-spread-function fitting and artificial-star recovery tests checked the photometry. The measurements require K corrections between received and emitted passbands and account for the observed $(1+z)$ stretching of light-curve times.^[ref]

Relevant systematic tests include dust and color laws, possible population evolution, flux-limited selection, calibration across filters and telescopes, gravitational lensing and host contamination. An accelerating interpretation is assessed by fitting a specified cosmological family and its nuisance parameters, checking those effects and comparing independent observations. The two principal discovery teams provided independent distant-supernova analyses, including the 42-event Supernova Cosmology Project result.^[ref]

At maximum expansion, $a_{\max }=A/k$. With $a_0=1$, the present Friedmann equation gives $k{c}^2=H_0^2({\Omega }_{M,0}-1)$, so positive curvature requires ${\Omega }_{M,0}>1$. Taking $H_0>0$ for the presently expanding branch givesThe lifetime of a closed dust universe follows by setting $\theta =2\pi$. The deceleration parameter follows from the Friedmann acceleration equation, $\ddot{a}/a=-4\pi G{\rho }_M/3$. Thus this model always decelerates while expanding. Its total lifetime includes the elapsed age; it is not the remaining lifetime.

The supernova discovery of accelerating expansion does not support this dust-only recollapse model as a description of the observed expansion history. Already the 1998 and 1999 Type Ia supernova analyses favoured accelerating expansion, requiring ingredients beyond this zero-cosmological constant, positive-curvature dust model within the usual FLRW metric interpretation.^[ref][ref]

Use standardized Type Ia supernovae as distance indicators and measure their redshifts. The curvature of the luminosity-distance relation constrains the deceleration parameter:Fainter supernovae at a given redshift imply larger distances, and comparison with the predicted curve tests accelerated expansion. In practice one fits the full expansion history over the observed redshift range and controls luminosity evolution, extinction and calibration, rather than relying only on a short low-$z$ series.

The cosmological fate from local expansion data is conditional on a matter model. For the spatially flat model of part b, if the only relevant components are pressureless matter and a constant $\Lambda$, put $A_0=\ddot{R}(t_0)/R(t_0)$. The two Friedmann equations giveHere $\Lambda$ follows the sign of the supplied Friedmann equations, where positive $\Lambda$ contributes positive acceleration. These fix the parameters governing subsequent evolution. Positive $\Lambda$ yields eternal late-time accelerated expansion; zero $\Lambda$ gives the flat matter solution; negative $\Lambda$ leads to eventual recollapse in this dust model. If instead one assumes dust and zero $\Lambda$ while allowing curvature, $q_0={\Omega }_{m0}/2$ determines whether the matter universe is closed and recollapsing or expands indefinitely.

Without a fixed equation of state and its future evolution, two present derivatives do not uniquely fix the future. For a concrete counterexample in units with $t_0=0$, the positive scale factors $a_{+}(t)=1+t+t^2/2$ and $a_{-}(t)=1+t+t^2/2-t^3$ have the same $a,\dot{a},\ddot{a}$ now. The former expands forever; the latter turns around and later reaches zero. Different time-dependent matter stresses can support their different flat expansion histories. Thus the printed sufficiency statement needs the cosmological-model assumption.

Each separately conserved component obeys the constant-equation-of-state density scaling $\rho \propto a^{-3(1+w)}$. For the flat Friedmann-Lemaître-Robertson-Walker metric, the Friedmann equations therefore giveWrite $H(z)/H_0=1+h_{1}z+O(z^2)$, where $h_1=[3{\Omega }_m+4{\Omega }_r+3(1+w){\Omega }_X]/2$. Expanding the reciprocal inside the luminosity distance integral givesMatching its quadratic coefficient gives the deceleration parameter $q_0=h_1-1$. Using flatness in this part,A negative deceleration parameter signifies accelerated expansion; negative pressure must overcome the positive matter and radiation terms. Restoring units replaces $H_0{d}_L$ by $H_0{d}_L/c$ in the expansion, leaving $q_0$ unchanged.

Type Ia supernovae have similar peak luminosities that can be standardized using their light-curve width and color, rather than assumed to be identical without correction. Their redshifts and corrected apparent magnitudes give a distance modulus,Compare this Hubble diagram with the luminosity-distance prediction of a background cosmology. Restoring $c$, put $E(z)=H(z)/H_0$, $I(z)={\int }_0^{z}d{z}^{\prime }/E(z^{\prime })$ and define ${\mathcal{S}}_{{\Omega }_k}(I)$ as $\sinh (\sqrt{{\Omega }_k}I)/\sqrt{{\Omega }_k}$ for ${\Omega }_k>0$, $I$ for ${\Omega }_k=0$, and $\sin (\sqrt{-{\Omega }_k}I)/\sqrt{-{\Omega }_k}$ for ${\Omega }_k<0$. ThenNeglecting radiation at these redshifts, Friedmann equations and a separately conserved dark energy component with $w(z)=p/(\rho c^2)$ giveFor constant $w$, the last factor is $(1+z{)}^{3(1+w)}$; a cosmological constant has $w=-1$. Thus the redshift dependence of the measured distance constrains matter density, curvature and the dark-energy equation of state. At small redshift, $d_L=(cz/H_0)[1+(1-q_0)z/2+\cdots ]$, showing how departures from the linear Hubble law probe the deceleration parameter.

The early distant-supernova analyses found larger distances than matter-only decelerating models predicted, providing evidence for accelerated expansion; see the 1998 supernova analysis and the 1999 independent supernova analysis. This is a statement about the distance-redshift curve under the standardization and cosmological assumptions, not a direct measurement of a fluid pressure.

The absolute magnitude–Hubble constant degeneracy means uncalibrated supernovae constrain the shape of $d_L(z)$ without separately determining $H_0$ and the common absolute luminosity. The integral over $H(z)$ also makes different matter, curvature and $w(z)$ histories partly degenerate. Wide redshift coverage and independent cosmic microwave background and large-scale structure of the universe information help break these degeneracies. Dust extinction, source evolution, photometric calibration, sample selection and K corrections must be modeled, and constraints on a time-dependent equation of state depend on how that function is parameterized. Standardized supernovae test the expansion history; they do not by themselves uniquely determine every cosmological parameter.

Use units $c=1$ and take the initially expanding branch with ordinary positive matter or radiation density. In the Friedmann-Lemaître-Robertson-Walker metric, $k=+1,0,-1$ respectively describe positive, zero and negative spatial curvature. Separately conserved pressureless matter has ${\rho }_M\propto a^{-3}$ and radiation in cosmology has ${\rho }_R\propto a^{-4}$, so either dominates the curvature and vacuum terms sufficiently far back on a hot Big Bang branch.

With negligible cosmological constant, a spatially flat radiation universe has $a\propto t^{1/2}$ and a flat matter universe has $a\propto t^{2/3}$; both expand forever while decelerating. An open universe also expands forever and eventually becomes approximately curvature dominated, with $a\propto t$. A closed universe containing only ordinary matter/radiation reaches a maximum size and recollapses. For example, dust gives ${\dot{a}}^2=C/a-k$ with $C>0$, so only $k=+1$ admits the positive turning point $a=C$.

Curvature alone does not determine the fate when the cosmological constant is retained. Positive $\Lambda$ can cause late accelerated expansion and prevent even a closed universe from recollapsing; whether a closed expanding branch turns around depends on the relative matter and vacuum terms. Negative $\Lambda$ can cause recollapse even for flat or open spatial sections. These statements follow by inspecting ${\dot{a}}^2=C/a-k+\Lambda a^2/3$ for dust; the curvature sign is a geometrical classification, not by itself a universal destiny criterion.

Define the critical density ${\rho }_c=3H^2/(8\pi G)$ and cosmological density parameters ${\Omega }_M={\rho }_M/{\rho }_c$, ${\Omega }_R={\rho }_R/{\rho }_c$, ${\Omega }_{\Lambda }=\Lambda /(3H^2)$ and ${\Omega }_k=-k/(a^2{H}^2)$. The Friedmann equation gives ${\Omega }_M+{\Omega }_R+{\Omega }_{\Lambda }+{\Omega }_k=1$. Dividing the Friedmann acceleration equation by $-H^2$ givesConsequentlyThe deceleration parameter measures the sign and size of the acceleration relative to $a{H}^2$: $q>0$ means deceleration, $q<0$ accelerated expansion and $q=0$ instantaneously zero acceleration. In particular a falling Hubble parameter alone does not imply deceleration, since $q=-1-\dot{H}/H^2$.

For conformal time $d\eta =dt/a$, the conformal Hubble parameter is $\mathcal{H}=aH$. Using matter conservation and differentiating ${\Omega }_M\propto {\rho }_M/H^2$,This is the matter density flow with a cosmological constant. Neglecting ${\Omega }_{\Lambda }$ during matter domination reduces it to the requested relationThat last equation is not exact with nonzero $\Lambda$. For example, a flat dust-plus-vacuum state with $({\Omega }_M,{\Omega }_{\Lambda })=(0.9,0.1)$ gives ${\Omega }_M^{\prime }/\mathcal{H}=-0.27$, whereas the matter-only expression gives $-0.09$.

For the flatness problem, let ${\Omega }_{\mathrm{tot}}={\Omega }_M+{\Omega }_R+{\Omega }_{\Lambda }$. Curvature satisfies ${\Omega }_{\mathrm{tot}}-1=k/(aH{)}^2$, soIn decelerating near-flat radiation and matter eras, deviations therefore grow as $a^2$ and $a$ respectively. Maintaining small curvature at a late epoch requires very small early deviations unless a prior mechanism suppresses them. Cosmic inflation, with $q\simeq -1$, drives the deviation down approximately as $a^{-2}$. If vacuum energy is present, flatness means ${\Omega }_{\mathrm{tot}}=1$, not ${\Omega }_M=1$.

For pressureless matter, the Friedmann acceleration equation with vanishing cosmological constant isEvaluating at the present cosmic time and using the definition of the deceleration parameter givesThe denominator is the critical density, so the cosmological density parameter satisfies the dust relationThis uses pressureless matter and $\Lambda =0$; spatial flatness has not been assumed.

For pressureless matter and zero cosmological constant, the Friedmann acceleration equation gives $\ddot{a}/a=-4\pi G\rho /3$. The deceleration parameter therefore satisfies the dust relation between deceleration and density, $q=4\pi G\rho /(3H^2)$. Evaluating today,Substitute this into the present Friedmann equation to obtain $H_0^2+k{c}^2/a_0^2=2q_0{H}_0^2$, orIn particular, $q_0>1/2$, $q_0=1/2$, and $q_0<1/2$ correspond to positive, zero, and negative spatial curvature.

Set $c=1$ and let $H=\dot{a}/a$ denote the proper-time Hubble parameter. The Friedmann equation is the expansion constraint: density drives expansion, while spatial curvature and the cosmological constant contribute separately. The Friedmann acceleration equation shows that the active gravitational density is $\rho +3P$; positive pressure gravitates, while positive vacuum energy accelerates expansion.

Subtracting the constraint from the acceleration equation gives $\dot{H}=-4\pi G(\rho +P)+k/a^2$. Differentiating the constraint gives $2H\dot{H}=(8\pi G/3)\dot{\rho }+2kH/a^2$. Eliminate $\dot{H}$ to obtainThis cosmological continuity equation is local energy conservation. For a comoving volume $V\propto a^3$, it is $d(\rho V)=-PdV$, the adiabatic first law: expansion does work against pressure.

Define the deceleration parameter $q=-a\ddot{a}/{\dot{a}}^2$, the critical density ${\rho }_c=3H^2/(8\pi G)$, and the cosmological density parameters ${\Omega }_m={\rho }_m/{\rho }_c$ and ${\Omega }_{\Lambda }=\Lambda /(3H^2)$. The dust-plus-vacuum acceleration equation then gives $q={\Omega }_m/2-{\Omega }_{\Lambda }$. Dust conservation gives ${\rho }_m\propto a^{-3}$, while ${\rho }_{\Lambda }=\Lambda /(8\pi G)$ is constant. At the acceleration transition, ${\rho }_m=2{\rho }_{\Lambda }$, so

For spatial flatness and $a_0=1$, the constraint today sets ${\Omega }_{\Lambda 0}=1-{\Omega }_{m0}$. Thus $H^2=H_0^2[{\Omega }_{m0}{a}^{-3}+1-{\Omega }_{m0}]$. Put $M={\Omega }_{m0}$, $L=1-M>0$ and $y=a^{3/2}$. Then $\dot{y}=(3H_0/2)\sqrt{M+L{y}^2}$. Integrating from $y(0)=0$ gives $\mathrm{arsinh}(y\sqrt{L/M})=(3H_0/2)\sqrt{L}t$, hence the flat dust-and-vacuum expansionFor small $t$, $a\simeq [(3/2)H_0\sqrt{M}t{]}^{2/3}$, the matter-dominated law. At late times, $a\simeq [M/(4L){]}^{1/3}{e}^{H_0\sqrt{L}t}$, approaching de Sitter spacetime. Negative spatial curvature adds a positive $a^{-2}$ contribution to $H^2$. This is subdominant to dust's $a^{-3}$ at small $a$ and to the constant vacuum term at large $a$, so it does not change either leading limit.

The model describes the transition from matter domination to the accelerated late-time expansion indicated by cosmological distance observations, with nearly flat geometry and vacuum-like dark energy. Its formal $t\to 0$ dust limit is not the actual earliest universe: radiation, neglected here, dominates then and gives $a\propto t^{1/2}$.

The particle horizon is the greatest distance from which a light signal could have reached the observer since the initial cosmic time $t_i$. Its comoving radius and its radial proper distance at time $t$ areHere proper distance is measured along the spatial slice, rather than by the transverse area of a sphere. The Hubble parameter, deceleration parameter and cosmological density parameter areThe last expression defines the critical density. For blackbody radiation, $p=\rho /3$, so the Friedmann acceleration equation gives $\ddot{a}/a=-\kappa \rho /3$. ConsequentlyThis is the radiation relation between deceleration and density. It does not require spatial flatness.

For the open radiation universe particle horizon, write $x=a/a_0$. The cosmological continuity equation gives $\rho ={\rho }_0{x}^{-4}$, while the present Friedmann equation gives $-k/a_0^2=H_0^2(1-{\Omega }_0)$. ThusTaking the big-bang endpoint $x=0$ and using $dt=dx/\dot{x}$, the present radial proper distance becomesThereforeThe formula applies to $0<{\Omega }_0<1$. Its flat-radiation limit is $H_0{d}_{h0}\to 1$, agreeing with $a\propto t^{1/2}$ and $d_h=2t=H^{-1}$. The divergence as ${\Omega }_0\to 0$ reflects the unbounded past conformal interval of the limiting empty open model; that endpoint is not a radiation-filled universe.

The horizon problem concerns the nearly uniform temperature of widely separated parts of the cosmic microwave background. In a purely decelerating Hot Big Bang model, their past light cones at last scattering do not overlap far enough to explain this agreement by thermal contact. During cosmic inflation, accelerated expansion shrinks the comoving Hubble radius. A patch initially small enough for causal communication can be stretched to encompass the later observable universe. Reheating converts the inflationary energy into a hot plasma with correlated initial conditions across that patch. Enough inflation must occur before the observable scales leave the Hubble radius; inflation cannot establish contact between regions that were never initially causally related.

The Friedmann acceleration equation and $H=\dot{a}/a$ give the deceleration parameterAn Einstein-de Sitter universe contains only pressureless matter, so $q_0=1/2$ and cannot explain the observed negative value. For the stated Lambda-CDM model,so its cosmological constant produces the required accelerating universe.

For radiation, $P=\rho c^2/3$ and $w=1/3$. The continuity equation gives $\rho \propto R^{-4}$, so the flat, zero-$\Lambda$ background expands as $R\propto t^{1/2}$. HenceRadiation pressure doubles the deceleration parameter relative to dust at the same density parameter. The printed radiation-filled Einstein-de Sitter wording is interpreted as a flat radiation-only model; the conventional name Einstein-de Sitter normally refers to the dust case.

For radiation in cosmology with zero cosmological constant, $p=\rho /3$. The Friedmann acceleration equation gives $\ddot{a}/a=-8\pi G\rho /3$, so the deceleration parameter equals the cosmological density parameter, regardless of spatial curvature. This is a relation for a pure radiation fluid, not a general identity for a mixture.

Stipulate $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ and $c=299792.458\mathrm{km}{\mathrm{s}}^{-1}$, so $c/H_0=4282.7494\mathrm{Mpc}$. A megaparsec is $10^6$ parsecs. Neglect radiation in cosmology. Compare an accelerating flat model $({\Omega }_{m,0},{\Omega }_{\Lambda ,0})=(0.3,0.7)$, an open matter-only model $(0.3,0)$ and the Einstein-de Sitter universe $(1,0)$. Their present deceleration parameters are respectively $-0.55$, $0.15$ and $0.5$. For these models,Here $\chi$ is dimensionless; the radial null geodesic in FLRW spacetime supplies its integral and spatial curvature of an FLRW universe supplies $S_k$. The observer-to-source comoving radial distance is $(c/H_0)\chi$, whereas the transverse distance uses $S_k$. Omitting the latter distinction would bias the open-model comparison.

At $z=0.5$, numerical integration gives $\chi =0.440984335$ for the flat accelerating model and $\chi =0.392227378$ for the open matter-only model. ThusThe accelerating model is $0.21559$ mag fainter than the open matter-only prediction at the same redshift and standardized absolute magnitude, corresponding to $1.10438$ times the luminosity distance and $0.81991$ times the radiative flux. These are model calculations, not observations from either collaboration.

The Milne model has $q=0$, ${\Omega }_{m,0}={\Omega }_{\Lambda ,0}=0$ and ${\Omega }_{k,0}=1$, giving $\chi =\log (1+z)$ and $d_L=(c/H_0)(z+z^2/2)$. The Einstein-de Sitter luminosity-distance relation independently checks the flat matter-only integral. At $z=0.5$, the accelerating model exceeds the Milne model by only $0.12317$ mag, illustrating the dependence of a residual on the chosen reference. Over a longer redshift interval, luminosity distance integrates past deceleration as well as recent acceleration; a residual need not increase forever.

Multiplying $H_0$ by $f>0$ at fixed density fractions divides every $d_L$ by $f$ and shifts every distance modulus by $-5{\log }_{10}f$. The predicted apparent magnitude is unchanged if $M_{B,0}$ simultaneously increases by $5{\log }_{10}f$. This is the absolute magnitude–Hubble constant degeneracy. For example, $70\to 77\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ shifts $\mu$ by $-0.20696$ mag and $M_{B,0}$ by $+0.20696$ mag. Relative curves at the same $H_0$ retain their separation; determining $H_0$ requires an external absolute magnitude calibration.